The influence of vegetation on turbulence and bed load transport

Authors

  • E. M. Yager,

    Corresponding author
    1. Center for Ecohydraulics Research, Department of Civil Engineering, University of Idaho, Boise, Idaho, USA
    • Corresponding author: E. M. Yager, Center for Ecohydraulics Research, Department of Civil Engineering, University of Idaho, Boise, ID 83702, USA. (eyager@uidaho.edu)

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  • M. W. Schmeeckle

    1. School of Geographical Sciences and Urban Planning, Arizona State University, Tempe, Arizona, USA
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Abstract

[1] Vegetation is ubiquitous in river channels and floodplains and alters mean flow conditions and turbulence. However, the effects of vegetation patches on near-bed turbulence, bed load transport rates, and sedimentation are not well understood. To elucidate the influence of emergent vegetation on local and patch-averaged bed load transport, we conducted a set of experiments in which we varied the mean flow velocity (U), total boundary shear stress (τ), or vegetation density between runs. We measured 2D velocity fields using Particle Imaging Velocimetry and bed load fluxes using high-speed video. Simulated rigid vegetation caused bed load fluxes to vary spatially by an order of magnitude, causing distinct scour zones adjacent to, and depositional bed forms between stems. These local patterns of sedimentation could impact recruitment and survival of other plants. Large bed load fluxes were collocated with high near-bed turbulence intensities that were three to four times larger than spatially averaged values. Higher vegetation densities increased the importance of inward and outward interactions, particularly downstream of vegetation. At the patch scale, greater stem densities caused either an increase or decrease in run-averaged bed load fluxes, depending on whether U or τ was held constant between runs. This implies that sedimentation in vegetation patches is not only a function of bed grain size, sediment supply, and vegetation density and species, but whether vegetation significantly impacts mean and local flow properties, which could depend on vegetation location. Commonly used bed load transport equations did not accurately predict average sediment fluxes in our experiments unless they accounted for the spatial variability in the near-bed Reynolds stress.

1 Introduction

[2] Vegetation is common on river bars and floodplains and can significantly alter local and reach-averaged flow, sediment transport rates, stream temperature, bank stability, and aquatic habitat [e.g., Ormerod et al., 1993; McKenney et al., 1995; Abernethy and Rutherfurd, 1998; Simon and Collison, 2002; Pollen, 2007]. Channel restoration projects often use vegetation to increase bank stability and provide shade and refugia for aquatic organisms. An understanding of the effects of vegetation on flow and sediment transport is essential for effectively restoring degraded channel conditions. Furthermore, vegetation could influence patterns of floodplain deposition and erosion, and channel form and migration rates [Murray and Paola, 2003; Allmendinger et al., 2005; Coulthard, 2005; Tal and Paola, 2007; Braudrick et al., 2009].

[3] The impact of vegetation typically varies with the flow properties (e.g., mean flow velocity, discharge, shear stress), and the vegetation type (e.g., rigid, flexible) and density by area [e.g., Carollo et al., 2002; Poggi et al., 2004; Ghisalberti and Nepf, 2006; Leonard and Croft, 2006; Bouma et al., 2007; Liu et al., 2008; Hopkinson and Wynn, 2009]. Vegetation creates velocity and turbulence intensity profiles that deviate from those commonly found in nonvegetated flows [e.g., Nepf, 1999; Leonard and Croft, 2006; Lightbody, 2006; Liu et al., 2008]. The exact velocity profile depends on the vegetation density and relative submergence [e.g., Nepf and Vivoni, 2000; Liu et al., 2008]. Vegetation also generates a variety of coherent flow structures that can significantly influence the pattern and magnitude of local and reach-scale flow turbulence. Large-scale structures consist of horseshoe vortices generated immediately upstream and adjacent to the vegetation [e.g., Liu et al., 2008; Stoesser et al., 2009], downstream von Karman vortices created by flow separation at the vegetation [e.g., Poggi et al., 2004], and Kelvin Helmholz structures [e.g., Poggi et al., 2004; White and Nepf, 2007] at the interface between vegetated and nonvegetated areas (both vertically and laterally). The generation of coherent flow structures can enhance turbulence, for example horseshoe vortices can increase local turbulence stresses by an order of magnitude [Escauriaza and Sotiropoulos, 2011]. Turbulence created in the stems wakes and by shear layers at vegetation patch edges also alter quadrant distributions (e.g., bursts, sweeps) of velocity fluctuations through the flow column [Poggi et al., 2004; Ghisalberti and Nepf, 2006; Yue et al., 2007; Nezu and Sanjou, 2008].

[4] Although flow patterns through vegetation have received considerable attention, few studies quantitatively examine the relationship between near-bed (<1 cm) flow turbulence and local and reach-averaged bed load transport. Previous investigations have related suspended sediment fluxes to changes in flow turbulence throughout the flow column but not at the bed [López and García, 1998; Zong and Nepf, 2011]. Enhanced suspended sediment deposition can occur within vegetation, particularly at the vegetation and open-channel interface [López and García, 1998; Cotton et al., 2006; Leonard and Croft, 2006; Sharpe and James, 2006; Zong and Nepf, 2011]. However, intervening areas between, or adjacent to vegetation patches may experience local erosion of suspended bed material. This is caused by high local velocities from constricted flow and the generation of vortices at the vegetation interface [Temmerman et al., 2007; Bouma et al., 2007; Rominger et al., 2010]. A key uncertainty is the spatial variability in near-bed turbulence and bed load transport rates within a vegetation patch. If a vegetation stem induces enhanced local erosion like a bridge pier, it may scour and uproot the plant [Fonseca et al., 1983; Rominger et al., 2010] whereas if local deposition is increased, the plant may be buried. Thus, the feedback processes between vegetation, near-bed turbulence, and bed load fluxes can influence future plant survival.

[5] Extensive research has been conducted on the parameters (e.g., pier diameter, bed grain size, flow depth) that influence bridge-pier scour depth [e.g., Melville, 1997; Ettema et al., 2006; Lee and Sturm, 2009], which is a similar process to that around a single rigid plant protruding through the flow depth. Recently, turbulence statistics within scour holes and the coherent flow structures generated by piers have been measured or modeled in detail [Dey and Raikar, 2007; Kirkil et al., 2008; Debnath et al., 2012; Kumar and Kothyari, 2012]. Such turbulence measurements are not necessarily representative of the conditions that occur in a patch of vegetation with many overlapping wakes. Few detailed measurements of bed load flux in the scour hole and on the surrounding bed exist, and these do not have concurrent near-bed turbulence data [Radice and Tran, 2012].

[6] In addition to stem scale processes, patch-averaged bed load fluxes have received relatively little attention. Vegetation patches are assumed to be associated with low sediment fluxes because they create higher drag and locally slower mean flow velocities than unvegetated areas experiencing the same flow discharge [e.g., Leonard et al., 2002; Shi and Hughes, 2002; Griffin et al., 2005; Cotton et al., 2006; Leonard and Croft, 2006]. Conversely, for the same patch-averaged flow velocity, a vegetation patch increases turbulence intensities [e.g., Nepf, 1999], which could cause greater sediment transport rates. Many other studies hold different parameters constant (e.g., flow depth, bed slope) and vary vegetation density or arrangement [Nezu and Onitsuka, 2001; Sharpe and James, 2006; Liu et al., 2008]. The observed influence of vegetation on flow turbulence highly depends on what is held constant between experiments or assumed constant in field comparisons. The influence of such experimental design on the presumed impact of vegetation on turbulence and sediment transport at the patch to reach scale is currently not known, which makes comparisons between studies difficult. Realistically, the addition of vegetation could alter the patch-averaged downstream velocity, flow depth, shear stress, and local discharge.

[7] In addition, many bed load transport equations use the excess shear stress in which the total shear stress is calculated as a function of depth and slope. However, Yager et al. [2007] demonstrated that use of the mean flow velocity instead of shear stress in bed load transport equations can result in better sediment flux predictions through large roughness elements. Recent studies have also demonstrated that the instantaneous downstream velocity, impulse, or force fluctuations are better correlated to sediment motion than reach-averaged shear stress [e.g.,Nelson et al., 1995; Schmeeckle et al., 2007; Diplas et al., 2008]. Therefore, a better understanding of the influence of mean flow velocity, shear stress, and turbulence fluctuations on bed load transport is needed, especially for the highly rough conditions that occur through vegetation patches.

[8] Here, we conduct detailed flume measurements of flow and bed load transport through rigid (e.g., trees or reeds) emergent vegetation. We use these data to answer the following questions: (1) what is the impact of vegetation density on patch-averaged near-bed turbulence, coherent flow structures and bed load fluxes, (2) what is the spatial variability in bed load flux and near-bed turbulence within vegetation patches, and (3) how accurate are reach-averaged calculations of bed load flux through vegetation?

2 Methods

[9] We conducted a set of 12 flume experiments in which uniform 0.5 mm sand was transported through staggered arrays (Figure 1) of 1.3 cm diameter emergent cylinders, which were used to simulate rigid vegetation (Table 1). We used rigid simulated vegetation to reduce the complexity of our experiments and to allow for detailed videos of flow and bed load transport (see below), which could not occur for flexible vegetation with stems and leaves. A large number of other flume experiments are based on similar assumptions [e.g., Poggi et al., 2004; Ghisalberti and Nepf, 2006; Bouma et al., 2007; Hopkinson and Wynn, 2009]. Our experiments have a ratio of vegetation diameter to grain size of 26, which scales to the transport of sand and fine gravel (e.g., 0.5–3.8 mm) through small trees or reeds (e.g., 1.2–10 cm stems [see McKenney et al., 1995]) in natural rivers.

Figure 1.

Photograph of simulated vegetation (cylinders) with transported sand. The box denotes the approximate area in which bed load fluxes were measured using a high-speed video camera. Lines denote locations of vertical PIV measurement transects.

Table 1. Experiment Runsa
Runa (%)SlopeQ (m3/s)h (m)U (m/s)τ (Pa)u* (m/s)Mean Bed Load Flux (cm2/s)ReFr
  1. aHere “a” is the vegetation density, τ is calculated as ρghS, and u* is determined from the average near-bed Reynolds stress. All parameters are run means, errors are standard errors. Standard errors for U range from 5×10−4 to 9×10−4 m/s, for h from 8×10−4 to 1×10−3 m and for bed load flux from 2×10−5 to 5×10−4 cm2/s. All standard errors are actually less than the accuracy of our measurements and are only shown to demonstrate their very low magnitudes. * denotes that there were likely large errors in the slope measurements when slope was very low, and we therefore estimated τ using the measured mean downstream velocity profile and the logarithmic velocity profile equation.
11.70.0029.9×10−30.200.174.40.0141.9×10−32.9 ×1040.12
21.70.0049.0×10−30.150.205.40.0101.7×10−22.6 ×1040.16
31.70.0051.5×10−20.200.259.80.0142.7×10−24.4 ×1040.18
400.001*1.3×10−20.180.250.23*0.0145.3×10−43.8 ×1040.19
50.80.0028.4×10−30.150.192.80.0102.1 ×10−32.5 ×1040.16
60.80.0031.2×10−20.130.294.40.0182.5×10−23.4 ×1040.26
70.80.0041.3×10−20.150.285.90.0066.4×10−33.8 ×1040.23
80.80.0047.9×10−30.110.234.00.0101.5×10−22.3 ×1040.22
940.0057.6×10−30.140.187.50.0121.6×10−22.2 ×1040.15
1040.0069.4×10−30.160.209.30.0034.7×10−22.8 ×1040.16
1140.0045.0×10−30.100.164.00.0011.2×10−21.5 ×1040.16
1240.0032.5×10−30.070.122.20.0097.1×10−47.4 ×1030.14

[10] We varied the simulated vegetation density by area (a), from 0 (no cylinders) to 4%, to mimic conditions observed in natural environments [see Nepf, 1999]. This density is calculated as the square of the ratio of the vegetation diameter to mean spacing. For a given a, we conducted one to four different runs in which we changed either the water discharge or the flume slope (Table 1). We did this to allow either the total boundary shear stress (τ) or the mean downstream flow velocity (U, average of all velocity measurements) to be held constant between runs with different a. We only conducted one run with a = 0 because sediment fluxes were zero or were too high to measure when we decreased or increased U (or τ), respectively, to similar values used for higher vegetation densities. We compared sediment fluxes for constant U or τ because of the following: (1) these are the parameters often used in bed load transport equations, (2) they are the readily available data in most rivers, (3) they are often held constant with changing roughness densities [e.g., Nepf, 1999; Lamb et al., 2008], and (4) correlations between time-averaged turbulence statistics and sediment flux are not always straightforward with nonuniform flow [Nelson et al., 1995].

[11] For a constant U or τ , the flow depth needed to vary with different a. Although turbulence intensities and coherent flow structures can vary with flow depth, our simulated vegetation was the dominant control of turbulence. For example, we partitioned the total shear stress between that borne by the sand bed and that borne by the simulated vegetation using the equations of Yager et al. [2007]. We substituted cylinders for spheres, assumed a drag coefficient for the cylinders ranging from 0.7 to 1.0 (depending on a [see Nepf, 1999]) and used our measured values of τ and U for each experiment (Table 1; see section 2.1) in these equations. The calculated stress borne by vegetation was always greater than or approximately equal to the measured total shear stress, which is not physically reasonable. Small uncertainties in flow measurements or drag coefficient estimates can produce large uncertainties in partitioned stresses if the vast majority of the total stress is borne by vegetation. Despite such uncertainties, these calculations demonstrate that the vegetation in our experiments controlled fluid momentum exchange and the effects of flow depth can be neglected. Similarly, bridge-pier scour depth, which is controlled by local turbulence, does not vary with flow depth if the downstream pier length is less than the flow depth [Melville, 1997], as is the case in our experiments.

[12] Before each experiment, a level, flat surface of sand was placed between cylinders and the water surface slope was adjusted to be parallel to the flume bed slope. All of the measurements were taken in the center of the flume length (8.5 m) and width (30.4 cm) to eliminate the effects of the entrance (flow baffles, no sediment feed), exit (tail gate), and near-wall conditions. The experiments were scaled to have Reynolds and Froude numbers representative of fully developed rough turbulent and subcritical flow (Table 1). Scour holes quickly developed around the vegetation, and all of our measurements were conducted after the scour holes stabilized. The duration of each run (three 20 s videos, see below) was relatively short to ensure that the upstream supply of sediment, from the bed to our measurement section, was not diminished.

2.1 Flow Measurements

[13] We measured the spatial and temporal variations in the downstream and vertical velocities using Particle Imaging Velocimetry (PIV). We seeded the flow with 60 micrometer neutrally buoyant particles and illuminated these particles in three downstream transects using a 4 W argon-ion laser sheet oriented parallel to the flow. A high-speed video camera recorded (from the side of the flume) the movements of the particles at 250 frames/s for approximately 19 s in each transect. The transects were located directly downstream and upstream of a cylinder and halfway between cylinder rows in the cross-stream direction (Figure 1). Each transect consisted of 74 vertical velocity profiles, which began at the bed surface, with 60 vertical points within a profile. The interrogation window for these spatial averages ranged from 0.08 cm by 0.08 cm to 0.12 cm by 0.12 cm, depending on the run. Most profiles occupied similar percentages of the flow depth and did not include measurements to the water surface because this exceeded the camera field of view. Our primary interest was near-bed flow conditions and not depth profiles, and we therefore focused our measurements in the lower portion of the flow column. Our measurement density was greater downstream and vertically than in the cross-stream direction. To adequately characterize the mean flow conditions in a run, our transect locations were positioned to represent the extremes in velocity that could occur. Additional transects would have been extremely time intensive for data processing, and our number of spatial measurement locations also far exceeds those in most studies on vegetation [e.g., Nezu and Onitsuka, 2001; Poggi et al., 2004; Liu et al., 2008; Nezu and Sanjou, 2008].

[14] We calculated the instantaneous downstream (u) and vertical (w) flow velocities within each interrogation region using the pixel shifts from a cross-correlation analysis between successive images [e.g., Adrian, 2005]. For each interrogation region, the fluctuating components of the downstream (u′) and vertical (w′) velocities are

display math(1)

where inline image and inline image are the time-averaged velocities. The downstream and vertical turbulence intensities (inline image) and Reynolds stresses (inline image) were calculated for each measurement point. Each flow parameter was then averaged though all measurement points to obtain run-averaged values. We also calculated the near-bed average of each of these parameters for every run. We defined near-bed flow conditions as those measured 0.5 cm from the sand-bed surface [Nelson et al., 1995]. Although we measured velocities closer than this, laser reflections on the sand surface sometimes precluded accurate velocity calculations below this level. In some locations (primarily for some of a =4%), the laser reflection height was greater than 0.5 cm above the sand surface and we chose the closest possible unimpacted near-bed points.

[15] We verified that our total number of flow measurements (~4912 for most experiments) at a given spatial location and our measurement duration were adequate (see Appendix A) to characterize turbulence statistics [see Buffin-Bélanger and Roy, 2005]. We did not correct our velocity coordinate system to reflect sloping bed surfaces near the vegetation because we did not measure the bed topography in detail. The lack of such of correction could influence the relative magnitudes of the downstream and vertical velocities but such errors should be relatively small.

[16] To better understand the impact of coherent flow structures on bed load transport, we performed quadrant analyses of the velocity fluctuations at each measurement point. We divided the joint distribution of u′ and w′ into four quadrants, bursts (u′ < 0, w′ > 0), sweeps (u′ > 0, w′ < 0), and inward (u′ < 0, w′ < 0) or outward (u′ > 0, w′ > 0) interactions. The joint distribution quadrant fractions (Si,H) are given by

display math(2)

where T is the total measurement duration, i is the quadrant (i.e., 1–4), H is the hole size, and x and z are the downstream and vertical coordinates, respectively [Lu and Willmarth, 2006]. The hole size excludes small magnitude momentum fluxes from the analyses and is often assumed to be 3–4 [Nakagawa and Nezu, 1977; Poggi et al., 2004]. We used two different values of H (0 and 3) to test the influence of the threshold value on our results. Ii,H is a conditional function that determines if a given measurement is within quadrant i and is given by

display math(3)

To compare the total proportion of the joint velocity distributions represented by each quadrant we calculated Pi,H,

display math(4)

where the sum of Pi,H for all i is one and each Pi,H is less than one. This calculation allows us to directly compare the change in proportions of different quadrants between different runs. Pi,H is a time-averaged quantity and was calculated for every measurement location in each run. We also calculated the near-bed spatially averaged values of Pi,H and Si,H for both each run and vegetation density (includes multiple runs). We calculated the ratio of average near-bed Pi,H values upstream of vegetation to those downstream, which quantifies spatial variations in Pi,H in each run. This analysis did not include cross-stream variations in flow velocities and did not allow us to determine the relative importance of larger coherent flow structures generated by vegetation. Although we are neglecting this 3-D flow variation in our calculations, we can still use quadrant analysis to determine the influence of vegetation on the occurrence of bursts, sweeps, and inward and outward interactions.

[17] In addition to turbulence statistics, we also calculated the run-averaged downstream velocity (U, average of all the time-averaged, downstream velocities in each experiment) and the total boundary shear stress (τ). The reach-averaged bed slope (S) was multiplied by the flow depth (h; average of 34–59 measurements along the flume walls in each run), the water density (ρ), and the acceleration due to gravity (g) to obtain τ (Table 1). In Run 4 (no vegetation), we calculated τ using the measured run-averaged downstream velocity profile and the logarithmic velocity profile equation because there could be relatively large errors in our slope measurements for this run. When the flume slope was very low, it was relatively difficult to measure in our experiments.

2.2 Sediment Transport Measurements

[18] We measured the spatial and temporal variations in sand transport at 250 frames/s using a high-speed video camera mounted above the flume (Figure 1). We visualized the bed through a window (smooth plastic sheet) that just touched the water surface and did not significantly impact the flow except immediately adjacent to the water surface. Local sediment transport rates were calculated using a fortran code that determined the difference in pixels (due to sand movement) though cross-correlation analyses between two successive video frames. At each measurement point (interrogation regions of similar sizes to those used in PIV), the total change in pixels was determined for the entire video duration in a run (about 40 s). Our analyses did not distinguish between individual and multiple grain movements (e.g., adjacent grains moved) and therefore required calibration to provide transport rates.

[19] We calibrated these pixel changes to visual counts of downstream grain transport at various sample measurement locations that coincided with the interrogation regions. We converted these grain counts to sediment transport rates (in cm2/s) by multiplying the total number of grains by the volume of a sand grain (assuming 0.5 mm diameter spherical grains) and then dividing by the measurement duration and the interrogation window width. We fit a relationship between the pixel shift and grain counts for a range of locations to obtain calibrated transport rates over several orders of magnitude (Figure 2). We then calculated the time-averaged sediment transport rate in every interrogation region using this calibration and the total local changes in pixels from the videos. The total number of grains (up to 1379) transported at a given location was adequate to characterize the time-averaged transport rate [see Roseberry et al., 2012]. Our calibration procedure relies on the fact that sediment flux is primarily modulated by the concentration of moving grains and to a much lesser extent by the average velocity of the sediment [e.g., Lajeunesse et al., 2010]. We calculated the dimensionless bed load flux as the measured bed load flux (in cm2/s) divided by the square root of the product of the gravitational acceleration, cubed sand radius, and ((ρs/ρ)−1) in which ρs is the sediment density.

Figure 2.

Calibration of sediment fluxes measured as the total pixel changes through all images in a given location and experiment. These values were correlated to total counts of individual sand grains at the same location.

[20] Our grain counts did not include vertical or cross-stream sediment motion, which could be especially significant immediately adjacent to the simulated vegetation [Radice and Tran, 2012]. Within the downstream scour hole, bed load recirculated, did not move uniformly downstream, and included significant vertical and upstream components of motion. Our calibration of bed load measurements did not include such effects and therefore may overestimate fluxes here. However, the scour holes visually had the highest downstream fluxes, and therefore the overall calibrated pattern of sediment transport was likely unaffected by this sediment recirculation.

2.3 Predictions of Bed Load Fluxes

[21] We predicted the run-averaged bed load flux in each of our experiments using the Fernandez Luque and Van Beek (FLVB) [Fernandez-Luque and Van Beek, 1976] and Parker [Parker et al., 1982] equations. We used a dimensionless critical shear stress of 0.045 [Buffington and Montgomery, 1997] in the FLVB equation. In each transport equation, we used three different representations of the near-bed applied shear stress for a given run. First, we used τ (ρghS), which includes vegetation drag and therefore overestimates the stress available to transport sediment. Second, we used the mean (of all measured values in a given run) near-bed Reynolds stress, which represents only the stress borne by sand and accounts for vegetation drag. Finally, we used every measured near-bed Reynolds stress, as a proxy for the actual skin-friction shear stress, to calculate local bed load transport rates and then averaged all these local fluxes for a given run. We did not use stress partitioning to calculate the stress borne by the sand because of the potential errors with this method highlighted above.

3 Results

[22] Although we briefly discuss the influence of vegetation on turbulence throughout the flow column, much of these analyses have already been performed by others. Our focus is on the linkages between near-bed flow conditions, coherent flow structures, and bed load transport rates and therefore most of our analyses use near-bed velocities.

3.1 Flow Patterns Around Vegetation

[23] We calculated run-averaged profiles using all measurements of a given parameter in the unvegetated run (Figure 3). We only show the average of 20 profiles, located approximately halfway between stems in the downstream direction, for each vegetated run. In a given vegetated run, the bed elevation at these locations was constant and allowed us to calculate an average profile at a set distance from the bed surface. Profiles immediately upstream or downstream of the vegetation usually had similar patterns in turbulence intensities or Reynolds stresses to what we show in Figure 3, but the magnitudes of these parameters were generally larger (see discussion below on Figure 4). We did not calculate run-averaged profiles for the vegetated runs because of the significant spatial variation in bed elevation that made defining consistent reference base levels for profiles problematic. Example approximate locations of near-bed values are shown in Figure 3, and measurements below this level are subject to error and therefore should be viewed with caution. The elevation of near-bed measurements was held constant at 0.5 cm from the bed, but the near-bed z + value varies because of different u* values between runs. In most runs, the near-bed measurements correspond to a z + of 40–60 (z/h of ~0.025–0.05), but a few runs had lower or higher values.

Figure 3.

Vertical profiles for different vegetation densities of the (a) downstream flow velocity normalized by the mean flow velocity (U), (b) downstream turbulence intensity/u*, (c) vertical turbulence intensity/u*, and (d) Reynolds stress/τ with an inset of detail for the vegetated runs. We defined u* as the square root of the run-averaged, near-bed (0.5 cm above the bed) Reynolds stress divided by the water density and defined τ as ρghS. The run without vegetation shows the average of all measured profiles whereas the vegetated runs each show the average of 20 profiles that were located approximately halfway between stems in the downstream direction. Three of the experiments (Runs 3, 4, and 8) were conducted with a similar mean flow velocity. z + denotes zu*/υ, where z is the distance above the bed and υ is the kinematic viscosity. Fits shown in Figures 3b and 3c are to measurements that were above ~0.2 z/h. Stars on each profile denote the locations of our near-bed values. Horizontal errors bars denote standard errors in the spatial mean at a given elevation. If error bars are not visible, they were either smaller than the width of the symbol/line or were not shown immediately adjacent to bed for the Reynolds stress to avoid obscuring many overlapping lines.

Figure 4.

The spatial variation in the normalized near-bed downstream flow velocity, downstream turbulence intensity, vertical turbulence intensity, and Reynolds stress (each parameter shown with a different line). Positive x locations are downstream of vegetation stems whereas negative x values are upstream and all data points are from transects directly in line with the stems. Each time-averaged parameter is normalized by its near-bed, time-, and space-averaged value and all data are for Run 1.

[24] The unvegetated run in Figure 3 was provided only as a reference to compare with the vegetated runs. For this unvegetated run, power functions fit (above ~0.15 z/h to correspond to previous studies) to the turbulence intensity profiles (Figure 3) had Du (2.9), Dw (1.6), λw (0.34), and λu (0.61) that were relatively similar to those reported in the literature (e.g., 2.04–2.3, 1.14–1.63, 0.67–1.0, and 0.88–1.0, respectively) [e.g., Nezu and Rodi, 1986; Kironoto and Graf, 1995; Debnath et al., 2012] although some discrepancies occurred. The normalized magnitudes of the near-bed downstream and vertical turbulence intensities were also very similar to those obtained in flume experiments and numerical simulations of fully developed, rough turbulent flow [e.g., Stoesser, 2010; Furbish and Schmeeckle, 2013]. The Reynolds stress in this unvegetated run had a peak value near the bed and then only declined slightly (Figure 3d) rather than showing the necessary linear decrease with higher vertical position for steady uniform flow. The Reynolds stress profile decline near the bed was partly because of the presence of a roughness sublayer (estimated elevation of z + of 12–18) and small amounts of sediment transport [Nikora and Goring, 2000; Bagherimiyab and Lemmin, 2013]. It should also be noted that Figure 3 only represents the bottom ~37% of the flow depth (vertical limit of our measurements), and the unvegetated run may not have reached steady, uniform flow at the downstream position of the measurements. However, our near-bed measurements were within the turbulence boundary layer that grows from the bed in a relatively short distance, and therefore our near-bed Reynolds stresses were representative of fully developed flow. For example, our run-averaged u* (0.014 m/s, calculated from the measured near-bed Reynolds stress) was very similar to that back calculated from the logarithmic velocity profile equation and measured flow velocities (0.015 m/s). In addition, the measured near-bed Reynolds stress corresponds to a Shields number of ~0.03 that implies sediment should be just at the onset of motion, which is supported by our visual observations and very low measured sediment fluxes in this run (Table 1).

[25] For the vegetated runs, the flow profiles were entirely dictated by the stems and bed topography and were independent of the flume entrance conditions. As expected, the addition of vegetation caused the run-averaged downstream velocity profile to significantly deviate from the logarithmic profile (Figure 3a). We scaled the turbulence intensities by u* (Figures 3b and 3c) to show the traditional expected scaling of the turbulence intensities profiles for the unvegetated run. We observed a slightly better collapse of the turbulence intensity profiles for all vegetated runs if we normalized by U. The average downstream and vertical turbulence intensities increased nearly everywhere in the flow column with vegetation addition (Figures 3b and 3c) and at all longitudinal locations (not shown here). Regardless of vegetation density, the greatest downstream turbulence intensities were near the bed. The highest vertical turbulence intensity without vegetation was near the bed whereas with vegetation there were two peaks, one near the bed and another in the middle to top of our measurements (z/h of 0.3–0.4). Relative to the unvegetated bed, vegetation addition significantly decreased the normalized Reynolds stresses throughout the flow profile. The vegetated beds often had negative stresses particularly in the middle of the measurements (Figure 3d). The significant variability in these Reynolds stress profiles was similar to that observed in numerical modeling calculations of flow around vegetation [Stoesser et al., 2009].

[26] In addition to vertical variability in the time-averaged flow conditions, near-bed turbulence parameters varied significantly with distance from vegetation. We use the near-bed flow conditions in Run 1 as examples here; the results in this run represent the general pattern of flow conditions in most other runs with vegetation. The downstream velocity declined to zero immediately upstream and downstream of the vegetation elements, as expected (Figure 4). The Reynolds stresses, and downstream and vertical turbulence intensities all reached peak values immediately adjacent to each stem. The downstream and vertical turbulence intensities were greatest immediately upstream (>2 times the mean) and downstream (>3 times the mean) of the vegetation, respectively. The Reynolds stresses reached peak positive values (~4 times the mean) upstream and peak negative values (~2 times the mean) downstream of the vegetation (Figure 4).

3.2 Quadrant Analyses

[27] We show spatial contour plots of Pi,H for the unvegetated run and for Run 1, which represents most other vegetated runs (Figure 5). The addition of vegetation significantly increased the spatial variability in Pi,H throughout the flow column and particularity near the bed (Figure 5). Bursts and sweeps were of greatest importance upstream of the vegetation, particularity within the lower half of the flow depth (Figures 5a and 5f). Inward and outward interactions were less frequent on average than bursts or sweeps but also displayed significant spatial variability in the vegetated runs (Figures 5a, 5b, 5e, and 5f). For example, inward interactions were highest near the bed but also appeared to be generated downstream of the vegetation in a plume that moved from the bed toward the water surface (Figure 5b).

Figure 5.

Contour plots of Pi,H, for (a, b, e, f) Run 1 (vegetation density of 1.7%) and (c, d, g, h) Run 4 (no vegetation). See Figure 3 for explanation of x-axis. The respective quadrants are labeled at the top right of each figure, and the sand bed is denoted by white areas without contours. Every figure uses the same contour interval and color scheme, which is shown in the legend. Flow is from left to right.

[28] If we focus only on near-bed events, sweeps and bursts dominated, and inward and outward interactions were relatively insignificant on a bed without vegetation (Figure 6). The addition of vegetation caused all near-bed Si,H (Figures 6a and 6b) to progressively increase primarily because inline image was relatively small (from the approximate balance of negative and positive values) at a given location but u′w′ for a given quadrant was large. The overall proportions (Pi,H) of near-bed sweeps and bursts declined, and inward and outward interactions became progressively more important with greater vegetation densities (Figure 6c). Further evidence for a decrease in the proportion of near-bed bursts and sweeps with vegetation addition is that near-bed Reynolds stresses were generally lower (more negative) on vegetated than unvegetated beds (Figure 3d). This is because bursts and sweeps are positive contributions to the Reynolds stress, and inward and outward interactions are negative contributions. Therefore, the Reynolds stress is small, and the contribution from each quadrant is relatively uniform with vegetation. In addition, the run-averaged Pearson Correlation Coefficient for the bed without vegetation was 0.43 (similar to near-bed values for smooth beds from Nakagawa and Nezu [1977] and Raupach [1981]), whereas for vegetated runs it ranged from 0.06 to 0.14 with an average value of 0.10. The low values for vegetated runs demonstrate that turbulence intensities had relatively poor correlation with Reynolds stress generation.

Figure 6.

Mean values of the (a, b) near-bed Si,H and (c) near-bed Pi,H for each quadrant (different symbols) as functions of the vegetation density. Hole sizes (H) of 0 (Figure 6a) and 3 (Figures 6b and 6c) are shown.

[29] The hole size (H) altered the absolute percentages of Si,H for a given vegetation density but did not change the overall pattern of results (Figures 6a and 6b). In quadrants three (inward interactions) and one (outward interactions), we also calculated the ratio of the average downstream to upstream (of vegetation) near-bed values of Pi,H in each run. These ratios ranged from 0.87 to 2.3 (mean of 1.3) for quadrant one and 0.95 to 3.0 (mean of 1.6) for quadrant three. Inward and outward interactions therefore generally were more important downstream rather than upstream of vegetation.

3.3 Constant Mean Downstream Velocity or Total Boundary Shear Stress

[30] We now focus on variations of run-averaged turbulence intensities and Reynolds stresses with U or τ. For the same U, vegetation addition generally caused higher run-averaged, near-bed downstream, and vertical turbulence intensities (Figures 7a and 7b). Only relatively small changes in the turbulence intensities occurred after the bed was already vegetated. In addition, the spatial variability of the near-bed turbulence intensities, as represented by the standard deviation, generally increased for a given U with higher vegetation densities (Figures 7c and 7d). The mean near-bed Reynolds stresses (Figure 7e) did not change systematically with vegetation addition, but were generally lower with vegetation than without (see section 3.2).

Figure 7.

The mean near-bed values for each run of (a) downstream turbulence intensity, (b) vertical turbulence intensity, (c) spatial standard deviation in the downstream turbulence intensity, (d) spatial standard deviation in the vertical turbulence intensity, and (e) Reynolds stress. Each parameter is plotted as a function of the mean (for a given run) downstream flow velocity (U), and different vegetation densities are shown with different symbols. Vertical standard error bars are shown for mean values unless they are smaller than the symbol size.

[31] For the same τ, vegetation addition did not produce systematic relationships between vegetation density and near-bed turbulence intensities (Figure 8). However, for many τ, an increase in vegetation density caused the downstream turbulence intensity to decline. For a given τ, the mean near-bed Reynolds stresses did not vary regularly with vegetation density and were generally smaller on vegetated than the unvegetated bed (Figure 8e). For a given vegetation density (except 0.8%), turbulence intensities increased with greater values of both U and τ, as expected (Figures 7 and 8).

Figure 8.

The mean near-bed values for each run of (a) downstream turbulence intensity, (b) vertical turbulence intensity, (c) spatial standard deviation in the downstream turbulence intensity, (d) spatial standard deviation in the vertical turbulence intensity, and (e) Reynolds stress. Each parameter is plotted as a function of the total boundary shear stress, and different vegetation densities are shown with different symbols. Vertical standard error bars are shown for mean values unless they are smaller than the symbol size.

3.4 Bed Load Transport

[32] To illustrate the spatial pattern of bed load transport around vegetation, we show contour plots of sediment fluxes from runs 3 (represents general pattern for vegetated runs) and 4 (no vegetation), which had the same U. Addition of vegetation significantly increased the spatial variability in bed load flux; the highest fluxes occurred immediately adjacent to, and downstream of vegetation with relatively low fluxes in the intervening area between vegetation stems (Figure 9). At the beginning of each run, net sediment deposition occurred in the areas between vegetation and scour holes developed adjacent to stems (Figures 10a and 10b). The depositional nonmigrating bed forms scaled with the distance between vegetation; the depositional area occupied almost the entire downstream distance between stems.

Figure 9.

Contour plots of bed load fluxes for (a) Run 3 (vegetation density of 1.7 %) and Run 4 (no vegetation). Flow is from right to left, and downstream (x-axis) and cross-stream (y-axis) distances are shown. The contour interval and color scheme for both figures are shown in the legend. Brown circles denote vegetation locations.

Figure 10.

Photographs of sediment scour and deposition for (a) a vegetation density of 1.7%, (b) a vegetation density of 0.8%, and (c) around real vegetation planted on a point bar in scaled outdoor flume experiments (at the Outdoor Stream Lab at Saint Anthony Falls Laboratory, University of Minnesota).

[33] We now examine the mean bed load transport rate in each run as a function of either U or τ. For the same U, an increase in vegetation density caused higher mean, and standard deviation in (spatial variability in the mean), bed load fluxes (Figures 11a and 11c). The differences in bed load fluxes between vegetated beds with different densities were much less than those that occurred between unvegetated and vegetated runs. The initial addition of vegetation caused about an order of magnitude increase in both mean flux and standard deviation in flux. Conversely, for the same τ, greater vegetation densities did not systemically impact the mean bed load fluxes (Figure 11b). For a given τ, with the exception of some runs with a vegetation density of 1.7%, the mean fluxes were roughly constant between different vegetation densities.

Figure 11.

The run-averaged dimensionless mean bed load fluxes for a range of vegetation densities (different symbols) as functions of the (a) mean downstream flow velocity (U), and (b) total boundary shear stress. (c) The spatial standard deviations in the bed load fluxes are also shown functions of U. Vertical standard error bars are shown unless they are smaller than the symbol size.

4 Discussion

4.1 Local Flow Turbulence, Bed Load Transport, and Sedimentation

[34] The spatial variations in bed load flux in our experiments are similar to those measured in a pressurized duct with a circular pier [Radice and Tran, 2012]. However, our calibrated bed load fluxes did not include cross-stream and vertical components of motion. The cross-stream component of sediment flux can be significant as sediment moves around cylinders [Radice and Tran, 2012] and then spreads laterally downstream. In our experiments, the measured large variability in downstream fluxes should be balanced by significant cross-stream and vertical components to produce zero divergence in bed load flux. Thus, our bed topography was stable despite large variability in the downstream bed load fluxes. This is similar to large variations in bed load fluxes measured over a stable alternate bar [Nelson et al., 2010].

[35] The spatial variability in bed load fluxes around vegetation should correlate to near-bed variations in flow turbulence [Nelson et al., 1995]. We could not simultaneously measure bed load transport rates and flow conditions and cannot correlate instantaneous sediment fluxes and flow velocities. We therefore focus on general time-averaged patterns of each parameter. High values of sediment flux, which occurred immediately adjacent to the vegetation, were generally collocated with areas of high near-bed Reynolds stresses and downstream and vertical turbulence intensities (Figures 4 and 9). The largest bed load fluxes and vertical turbulence intensities were located in the scour hole immediately downstream of the vegetation, whereas the absolute maxima of the Reynolds stresses and downstream turbulence intensities were approximately l cm upstream. The large bed load fluxes near vegetation were therefore likely caused by high turbulence intensities, possibly more so those in the vertical direction. Cross-stream velocities could also transport significant amounts of sediment because of their large magnitudes around vegetation [Hopkinson and Wynn, 2009; Stoesser et al., 2009].

[36] Sweeps and bursts were more important upstream of vegetation, and inward and outward interactions were greater downstream, both throughout the flow column (Figure 5) and particularly near the bed (section 3.2). The inward and outward interactions downstream of the vegetation may be from highly local vertical flow [Stoesser et al., 2009] and oscillating von Karman vortices [Poggi et al., 2004] produced by the vegetation. Outward interactions can transport more sediment than other events of the same magnitude and duration [Nelson et al., 1995] and may be partially responsible for high sediment fluxes downstream of vegetation. High, but not peak values, of sediment fluxes immediately upstream and adjacent to vegetation may be caused by horseshoe vortices, which can locally increase turbulence by an order of magnitude [Escauriaza and Sotiropoulos, 2011]. Low bed load fluxes between vegetation stems were collocated with reduced turbulence intensities, and sediment deposition here may be caused by von Karman vortices in a manner similar to deposition on the outsides of fast turbulent jets entering low velocity pools [Rowland et al., 2005]. Some of the measured changes in near-bed turbulence with vegetation addition could be from scour hole and depositional ridge development rather than from the vegetation directly but such topographic alterations were still caused by vegetation and we cannot quantify their independent effects.

[37] Our quadrant analyses results differ from other studies on vegetation patches or bridge pier scour in which bursts and sweeps dominate and inward and outward interactions are relatively insignificant. Most of these studies focused on turbulence events at the top of submerged vegetation [Poggi et al., 2004; Ghisalberti and Nepf, 2006;Yue et al., 2007; Nezu and Sanjou, 2008] and none measured near-bed conditions (≤0.5 cm). Our experiments lacked the shear layer caused by submerged vegetation, which induces greater frequencies of sweeps. Much of the total stress in our runs was from stem drag (section 2), and therefore relatively little vertical momentum transport in the form of bursts and sweeps was needed. This is supported by the low Pearson Correlation Coefficient in our vegetated runs. Nezu and Sanjou [2008] similarly observed that inward and outward interactions increased toward the bed but only found this for a relatively high vegetation density. Therefore, our study is the first to demonstrate that near-bed inward and outward interactions can be important for a range of vegetation densities.

[38] The scour holes immediately adjacent to stems and depositional bed forms in intervening areas (Figures 10a and 10b) are similar to those observed with bridge pier scour [e.g., Ettema et al., 2006] and pronating vegetation (see Figure 10c) in flume experiments [Coulthard, 2005; Rominger et al., 2010] and in the field [Leonard and Luther, 1995; Bouma et al., 2007]. This consistent topographic pattern may be important because vegetation stability will be reduced by local scour, depending on the relative scour depth and root strength [Fonseca et al., 1983; Rominger et al., 2010]. Furthermore, downstream deposition could bury and smother neighboring vegetation seedlings, resulting in a possible characteristic downstream separation between established vegetation. In such cases, we would expect to see a uniform stem pattern as opposed to a clustered or completely spatially random pattern. The increases in topographic complexity caused by vegetation could also lead to greater species richness [Pollock et al., 1998].

[39] We do not imply that these exact topographic patterns will be observed around all vegetation species. Specific sedimentation patterns are highly influenced by the upstream sediment supply (quantity and grain size), location of vegetation within the channel, vegetation species (diameter, presence of stems and leaves, pronation, height) and density, and the grain sizes of the bed. Wide distributions of sand and/or gravel sizes may result in the formation of coarse or fine sediment patches in zones of spatially increasing (positive divergence) or decreasing (negative divergence) shear stress, respectively [Nelson et al., 2010]. Such grain size variability could alter the spatial patterns in bed load volumes and sedimentation in our experiments. For example, the scour depth around bridge piers is inversely proportional to the ratio of bed grain size to bridge pier diameter [Lee and Sturm, 2009]. This occurs because coarser grains are usually more difficult to erode (higher critical shear stresses) than fine sediment, whereas larger obstructions can generate greater local turbulence to cause motion. The elucidation of such complex local feedback interactions between flow turbulence, grain size, sedimentation, and vegetation will require future flume experiments or numerical modeling. We hypothesize that calculations of vegetation stability that neglect any of these factors could result in erroneous predictions of vegetation survival.

4.2 Patch-and Reach-Scale Flow Turbulence, Bed Load Transport, and Sedimentation

[40] Higher vegetation densities caused mean bed load fluxes to increase for a given U (Figure 11) because vegetation addition increased run-averaged near-bed turbulence intensities (Figure 7), which has also been observed for turbulence throughout the flow column [e.g., Nepf, 1999]. Conversely, for constant τ, vegetation addition caused relatively small changes in mean bed load fluxes (Figure 11) because of either no systematic change or a slight decrease in the near-bed turbulence intensities (Figure 8). Such decreases in turbulence are commonly observed when the shear velocity is held constant and roughness (boulders, steps, etc.) is increased [Sumer et al., 2003; Lamb et al., 2008]. Thus, depending if U or τ is constant, greater vegetation densities will either increase or not significantly influence bed load fluxes because of the respective changes in near-bed turbulence intensities and possibly mean flow conditions.

[41] This result is important for the following reasons: (1) the actual impact of vegetation on bed load fluxes and sedimentation will depend on how vegetation alters patch-averaged flow conditions and (2) this influence on the mean flow likely varies with vegetation position within a channel. In the introduction, we demonstrated that there is not one parameter that is consistently held constant to assess vegetation (or other roughness) addition effects, which makes generalizations difficult. In real rivers, few patch-averaged parameters are constant with increased vegetation density; vegetation can often decrease U and local discharge, and increase h and τ. In particular, vegetation position has the potential to significantly impact sediment transport if patch-averaged U, τ, or neither is constant [Green, 2005a].

[42] U can remain relatively constant when flow first enters a midchannel vegetation patch because the initial vegetation front is not sufficient to significantly reduce high velocities here [Green, 2005b] (see figures in Bouma et al. [2007]). Thus, the addition of vegetation to a midchannel bar could increase turbulence intensities and bed load fluxes (Figures 7 and 11), which if not balanced by the sediment supply, may cause scour at the upstream end of the bar. This hypothesis is generally supported by observed scour at vegetation patch edges that are subjected to high flow velocities and the generation of coherent flow structures [Bouma et al., 2007; Temmerman et al., 2007; Rominger et al., 2010]. Caution may therefore be warranted in using vegetation as a restoration measure on channel bars. Conversely, vegetated patches on the channel margins, banks, or at the downstream end of a midchannel bar can decrease local τ and/or U near the patch edge [Griffin et al., 2005; White and Nepf, 2008; Hopkinson and Wynn, 2009]. For these locations, sediment fluxes could decrease through the vegetation, and assuming a sufficient upstream sediment supply, deposition could occur. Additional flow modeling and/or field measurements of these parameters (U, bed load flux, etc.) with and without vegetation are needed to determine the optimal vegetation location(s) for different river restoration goals. The influence of vegetation location on bed load fluxes could have larger-scale impacts on channel form, migration rates, and width as well as long-term sedimentation rates and processes.

4.3 Predictions of Sediment Flux Through Vegetation

[43] Use of the total shear stress in either the FLVB or Parker equations systematically (except for the unvegetated run) overpredicted mean bed load fluxes by several orders of magnitude because a large portion of this stress was borne by the vegetation (Figure 12). Conversely, use of the average Reynolds stress in either equation systematically under-predicted bed load fluxes by several orders of magnitude for 100% of our runs. In particular, 25% and 91% of our experiments essentially had a predicted flux of zero (defined as less than 1×10−10) for the Parker and FLVB equations, respectively. Use of the Reynolds stress spatial distribution caused the predicted mean bed load fluxes to be within an order of magnitude of the measured values for 70 and 50% of the data points (Figure 12) for the FLVB and Parker equations, respectively. Such accuracy is considered relatively good for channels with significant roughness [Yager et al., 2007].

Figure 12.

The log of the ratio of the predicted to measured sediment volumes for different sediment transport equation (Parker or FLVB) and shear stress combinations. Values were calculated using the total boundary shear stress (τ), mean near-bed Reynolds stress (Re with overbar), and distribution of near-bed Reynolds stresses (Re). All 12 experiments are included in the box plots, and no systematic difference in transport prediction accuracy was observed with different a. Positive and negative volume ratios are predictions that were greater or less than, respectively, the measured values. The median is the line in the middle of each box, and 25th and 75th percentiles of the data points are the top and bottom of each box, respectively. Lines extending out of each box denote the furthest limit of the data points but only include ratios greater than −1×10−10. Ratios less than this value were when the predicted sediment transport was either 0 or over 10 orders of magnitude less than the measured value; the number of runs with such ratios are denoted next to the black arrows. The solid and dashed horizontal gray lines are predicted fluxes equal to or within an order of magnitude of the measured volumes, respectively.

[44] The distribution of near-bed Reynolds stresses was needed to obtain relatively accurate fluxes because bed load transport is a nonlinear (1.5 power) function of the excess shear stress (skin-friction shear stress–critical shear stress). Thus, small changes in the skin-friction shear stress can yield large differences in predicted sediment fluxes. Local measurements or simulated values of the Reynolds stress may therefore be needed to obtain accurate mean bed load fluxes through vegetated areas. Caution is needed when predicting bed load fluxes, even through relatively low vegetation densities, if such detailed data are not available. In this analysis, we do not suggest that the near-bed Reynolds stress is the flow parameter responsible for sediment motion. Rather, whatever flow parameter is used in a standard bed load transport equation will need to be applied as a spatial distribution function rather than a single mean value. Indeed, our predicted bed load fluxes are still not highly accurate even using the measured Reynolds stress spatial distribution. Simple drag law arguments suggest that the near-bed mean velocity along with the turbulence intensity may provide better predictions of bed load flux in future studies.

5 Conclusions

[45] Bed load flux within a vegetation patch can be highly spatially variable, because of variations in the near-bed turbulence intensities and coherent flow structures, and result in distinct scour holes and depositional features. At the patch scale, vegetation can either enhance or have little impact on mean bed load fluxes depending on whether the mean downstream flow velocity (U) or total boundary shear stress (τ) is held constant. Therefore, the influence of vegetation on bed load transport not only depends on its density and type but how it alters patch-averaged flow conditions. Caution is warranted in assuming either local or patch-averaged vegetation impacts on sedimentation rates because the influence of vegetation on sediment fluxes can be highly variable. The large variability in bed load fluxes within a vegetation patch decrease the accuracy of commonly used bed load transport equations. The spatial distribution of the near-bed Reynolds stress may be needed to obtain reasonably accurate sediment flux predictions though vegetation. Additional research on the combined influence of vegetation type (e.g., flexible vs. rigid) and patch location, bed grain size, and upstream sediment supply is needed to improve bed load flux predictions, river restoration design, and possibly calculations of long- and short-term sedimentation rates.

Appendix A

[46] To adequately characterize turbulence statistics, a large number of measurements and a measurement duration of sufficient length are needed to include both high- and low-frequency turbulence fluctuations. To demonstrate that our measurement duration was adequate, we calculated running averages of the instantaneous downstream velocity, Reynolds stress, and joint distribution quadrant fractions at a near-bed location (Figure A1). These values stabilized after ~2000–3000 frames (8–12 s) in most locations for each run, which is less than our measurement duration.

Figure A1.

Running averages of (a) u, (b) u′w′, and (c) joint distribution quadrant fractions at a location 0.5 cm from the bed in Run 5.

Acknowledgments

[47] Support for this research was provided by NSF grant EAR 0352079 to M. Schmeeckle. The authors acknowledge helpful comments from three anonymous reviewers, AE, and Editor A. Densmore.

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